By Vladimir Mazalov, Dmitry Novikov, Guennady Ougolnitsky, Leon Petrosjan
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Additional resources for Game Theory and Applications: Game-theoretic Models in Mathematical Ecology
18) The Principle of Coordinated Planning in Control … 35 Remark. By assumption, for each agent Qj(k) ≠ and, moreover, there exist projects with a guaranteed effect to the agent. If not, an agent should design projects with a sufficiently high effect. Example 1. Take 2 subprograms and 2 agents, each having 4 projects. The corresponding parameters are combined in Table 27. Table 27. 8 Suppose that projects 1-4 (projects 5-8) are candidates for inclusion in subprogram 1 (subprogram 2, respectively).
X X div(UX ) X X (v X ) X S XS t z z X XZ X X X XP; Z Z div( UZ ) Z Z (vZ ) Z X XZ Z Z Z Z ; t z z S S div( US ) S S (vS ) S D D S X B( S p S ) f ; t z z (5) D D div( UD) D D (vD ) X X Z D D D DP; t z z P P div(U P P) p P (vP ) P D DP P P t z z P X XP P P uP; u P u div(U Pu P ) u u P (vu P ) u u P t z z k D gradD k X gradX.
Obviously, C(v) for sufficiently large and -core is a subset of the set of preimputations (preimputation is a vector satisfying only the condition of collective rationality). It has the property of stability under the assumption that it is necessary to make the costs equal for the creation of any coalition. If the original С, but too wide, it can be narrowed by the selection < 0, which is then interpreted as a reward for the creation of a coalition. The least-core of the game is the intersection of all nonempty -cores.
Game Theory and Applications: Game-theoretic Models in Mathematical Ecology by Vladimir Mazalov, Dmitry Novikov, Guennady Ougolnitsky, Leon Petrosjan